Tsujikawa, Phys. Rev. D 77, 023507 (2008)

Equation of state for dark energyin modified gravity theoriesMain references:[1] K. Bamba, C. Q. Geng and C. C. Lee,JCAP 1011, 001 (2010) [arXiv:1007.0482 [astro-ph.CO]].[2] K. Bamba, C. Q. Geng, C. C. Lee and L. W. Luo,JCAP 1101, 021 (2011) [arXiv:1011.0508 [astro-ph.CO]].[3] K. Bamba, S. Nojiri, S. D. Odintsov and M. Sasaki, arXiv:1104.2692 [hep-th].KMI Inauguration Conference on ``Quest for the Origin ofParticles and the Universe'' (KMIIN) on October 24, 2011ES Hall, Engineering Science (ES) Building (KMI site),Nagoya University, NagoyaPresenter : Kazuharu Bamba (KMI, Nagoya University)Collaborators :Chao-Qiang Geng, Chung-Chi Lee, Ling-Wei Luo (NationalTsing Hua University), Shin'ichi Nojiri (KMI and Dep. ofPhysics, Nagoya University), Sergei D. Odintsov (ICREA andIEEC-CSIC), Misao Sasaki (YITP, Kyoto University and KIAS)

I. IntroductionNo. 3・ Recent observations of Supernova (SN) Ia confirmed thatthe current expansion of the universe is accelerating.[Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517,565 (1999)][Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, 1009 (1998)][Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]・ There are two approaches to explain the current cosmicacceleration. [Copeland, Sami and Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)][Tsujikawa, arXiv:1004.1493 [astro-ph.CO]]< Gravitational field equation > G ö÷G ö÷ = ô 2 T ö÷GravityMatter(1) General relativistic approachT ö÷(2) Extension of gravitational theory2011 Nobel Prize in Physics: Einstein tensor: Energy-momentum tensor: Planck massDark Energy

(1) General relativistic approach・ Cosmological constant・ Scalar fields:X matter, Quintessence, Phantom, K-essence,Tachyon. f(R)・ Fluid: Chaplygin gas(2) Extension of gravitational theory・ f(R) gravity・ Scalar-tensor theories・ Ghost condensates: Arbitrary function of the Ricci scalar[Capozziello, Cardone, Carloniand Troisi, Int. J. Mod. Phys. D12, 1969 (2003)][Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70,043528 (2004)][Nojiri and Odintsov, Phys. Rev. D 68, 123512 (2003)]R[Arkani-Hamed, Cheng, Luty and Mukohyama,JHEP 0405, 074 (2004)]G : Gauss-Bonnet termf(G) T : torsion scalar[Dvali, Gabadadze and Porrati, Phys.Lett B 485, 208 (2000)]・ Higher-order curvature term ・ gravity・ DGP braneworld scenario・ Non-local gravity [Deser and Woodard, Phys. Rev. Lett. 99, 111301 (2007)]・ f(T) gravity[Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)]・ Galileon gravity[Linder, Phys. Rev. D 81, 127301 (2010) [Erratum-ibid. D 82, 109902 (2010)]][Nicolis, Rattazzi and Trincherini, Phys. Rev. D 79,064036 (2009)]No. 4

Flat Friedmann-Lema tre-Robertson-Walker (FLRW)space-time >a (t)No. 5: Scale factor< Equation for a(t) with a perfect fluid >ä = àô 2 ( 1+3w)úa 6w ñ úP: Equation of state (EoS)T ö÷ =diag(ú, P, P, P)ú : Energy densityP : Pressureä > 0 : Accelerated expansionw

m à:mMMDistanceestimator::ApparentmagnitudeAbsolutemagnitude< SNLS data >Pure mattercosmologyΩ m =1.00Ω Λ =0.00zNo. 6Flat ΛcosmologyΩ m =0.26Ω Λ =0.741H 2 0ä Ω=aà m2(1 + z) 3 + Ω ΛôΩ m ñ 2 ú(t 0 )3H20ΛΩ Λ ñ 3H20: Density parameter for matter: Density parameter for ΛFrom [Astier et al. [The SNLSCollaboration], Astron. Astrophys.447, 31 (2006)]1+z = aa 0, za 0 =1: Red shift‘‘0’’ denotes quantitiesat the present time . t 0

7-year WMAP data on the current value of >K =0・ For the flat universe, constant w :Cf.Ω Λ =(68% CL)(68% CL)(95% CL)(FromwKΩ K ñ (a0 H 0 ) 2No. 7From [E. Komatsu et al. [WMAPCollaboration], Astrophys. J. Suppl.192, 18 (2011) [arXiv:1001.4538[astro-ph.CO]]].Hubble constant ( H 0 ) measurementBaryon acoustic oscillation (BAO): Special pattern in the large-scalecorrelation function of SloanDigital Sky Survey (SDSS)luminous red galaxies: Time delay distance: Density parameterfor the curvatureK =0 : Flat universe.)

Canonical scalar field >⎧S þ = ⎭ d 4 √x à gâà 1 g ö÷ ã∂ 2 ö þ∂ ÷ þ à V(þ)No. 9g =det(g ö÷ )V(þ)þ : Scalar field: Potential of þ・ For a homogeneous scalar field þ = þ(t) :ú þ = 2 1 þ ç 2 + V(þ), P þ = 2 1 þ ç 2 à V(þ)Pw þ = þ þ= ç 2 à2V(þ)úþ þ ç 2 +2V(þ)þ ç 2 ü V(þ)w þ ùà1If , .Accelerated expansion can be realized.

f(R) gravity >No. 10Sf(R)2ô 2f(R) gravityf(R)=R: General Relativity[Nojiri and Odintsov, Phys. Rept. 505, 59 (2011) [arXiv:1011.0544 [gr-qc]];Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007) [arXiv:hep-th/0601213]][Capozziello and Francaviglia, Gen. Rel. Grav. 40, 357 (2008)][Sotiriou and Faraoni, Rev. Mod. Phys. 82, 451 (2010)][De Felice and Tsujikawa, Living Rev. Rel. 13, 3 (2010)]< Gravitational field equation >f 0 (R)=df(R)/dR: Covariant d'Alembertian: Covariant derivative operator

・In the flat FLRW background, gravitational field equations readú eff , p eff,: Effective energydensity andNo. 11pressure from theterm f(R) à RExample:öf(R)=R à 2(n+1)Rn[Carroll, Duvvuri, Trodden and Turner,Phys. Rev. D 70, 043528 (2004)]・ö : Mass scale, n : Constant Second term becomea ∝ t q (2n+1)(n+1), q =important as R decreases.n+2If q>1 , accelerated2(n+2)w eff = à 1+ 3(2n+1)(n+1) expansion can be realized.(For n =1, q =2 and w eff = à 2/3 .)

Conditions for the viability of f(R) gravity >(1) f 0 (R) > 0f 0 (R) ñ df(R)/dR(2)(3)f 00 (R) > 0f 00 (R) ñ d 2 f(R)/dR 2(4) 0 0 GM 2 ù 1/(3f 00 (R)) > 0[Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)][Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)][Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)][Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)](5) Constraints from the violation of the equivalence principle(Solar-system constraints) [Chiba, Phys. Lett. B 575, 1 (2003)][Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]Scale-dependence ‘‘Chameleon mechanism’’M = M(R) :Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)]No. 12Positivity of the effective gravitational couplingStability condition:: Gravitational constantM :Mass of a new scalar degree of freedom (“scalaron”)in the weak-field regime.Existence of a matterdominatedstagef(R) → R à 2Λ for R ý R 0 .R 0 : Current curvature, Λ : Cosmological constant Stability of the latetimede Sitter pointCf. For general relativity,m =0.

Models of f(R) gravity (examples) >[Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)](i) Hu-Sawicki modelCf.No. 13[Nojiri and Odintsov, Phys. Lett. B 657, 238(2007); Phys. Rev. D 77, 026007 (2008)]: Constant parameters(ii) Starobinsky’s model [Starobinsky, JETP Lett. 86, 157 (2007)]: Constant parameters(iii) Tsujikawa’s model [Tsujikawa, Phys. Rev. D 77, 023507 (2008)](iv) Exponential gravity model: Constant parameters[Cognola, Elizalde, Nojiri, Odintsov,Sebastiani and Zerbini, Phys. Rev. D 77,046009 (2008)][Linder, Phys. Rev. D 80, 123528 (2009)]: Constant parameters

Crossing of the phantom divide >・ Various observational data (SN, Cosmic microwavebackground radiation (CMB), BAO) imply that the EoS ofdark energy w DE may evolve from larger than -1 (nonphantomphase) to less than -1 (phantom phase). Namely, itcrosses -1 (the crossing of the phantom divide).w DEà 1z c0zz cRedshift:z ñ 1 aà 1: Red shift at the crossingof the phantom divideNo. 14[Alam, Sahni and Starobinsky, JCAP 0406, 008 (2004)][Nesseris and Perivolaropoulos, JCAP 0701, 018 (2007)][Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)](a)w DE > à 1Non-phantom phase(b) w DE = à 1Crossing of the phantom divide(c) w DE < à 1Phantom phase

Data fitting of w(z) >zw(z)=w 0 + w 1 1+zNo. 15From [Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)].SN gold data set[Riess et al. [Supernova SearchTeam Collaboration],Astrophys. J. 607, 665 (2004)]SNLS data set[Astier et al. [The SNLSCollaboration], Astron.Astrophys. 447, 31 (2006)]Shaded regionshows 1û error.+Cosmic microwave background radiation (CMB) data[Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 170, 377 (2007)]SDSS baryon acoustic peak (BAO) data[Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)]

・ It is known that in several viable f(R) gravity models,the crossing of the phantom divide can occur in thepast.(i) Hu-Sawicki model[Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)][Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, 123516 (2009)]Cf. [Nozari and Azizi, Phys. Lett. B 680, 205 (2009)](ii) Starobinsky’s model(iv) Exponential gravity modelCf.[Motohashi, Starobinsky and Yokoyama, Prog. Theor.Phys. 123, 887 (2010); Prog. Theor. Phys. 124, 541 (2010)][Linder, Phys. Rev. D 80, 123528 (2009)][KB, Geng and Lee, JCAP 1008, 021 (2010) arXiv:1005.4574 [astro-ph.CO]]Appleby-Battye modelNo. 16[Appleby, Battye and Starobinsky, JCAP 1006, 005 (2010)]

w DEw DE = à 1< Cosmological evolution of in the exponentialgravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)].No. 17w DE (z =0)=à 0.93(< à 1/3)f E (R)=Crossingin the pastCrossing of thephantom divide

We explicitly demonstrate that the future crossingsof the phantom divide line w DE = à 1 are thegeneric feature in the existing viable f(R) gravitymodels.No. 18・Recent related study on the future crossings of thephantom divide:[Motohashi, Starobinsky and Yokoyama, JCAP 1106, 006 (2011)]

II. Future crossing of the phantom divide in f(R) gravity< Action > g =det(g ö÷ )f:: Metric tensorArbitrary functionofRNo. 19: Action of matter< Gravitational field equation >: Matter fields: Energy-momentum tensor of allperfect fluids of matterR ö÷: Ricci tensor: Covariant derivative operator: Covariant d'Alembertian

Gravitational field equations in the FLRW background:No. 20ú M: Hubble parameterand P M : Energy density and pressure of all perfectfluids of matter, respectively.Pw DE ñ DEúDE・Analysis method: [Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)]

Future evolutions of 1+w DE as functions of z >No. 21Exponential gravity model1+w DECrossings in the future01+w DE =0Crossing ofthe phantomdivideRedshift:z ñ a 1 à 1( z

Hu-Sawicki modelStarobinsky’s modelNo. 221+w DERedshift:z ñ a 1 à 1( z

III. Equation of state for dark energy in f(T) theoryNo. 23e A (x ö ) : Orthonormal tetrad componentsAn index A runs over 0, 1, 2, 3 for: Torsion tensor the tangent space at each point of x öthe manifold.ö and ÷ are coordinate indices on the: Contorsion tensormanifold and also run over 0, 1, 2, 3,and e A (x ö ) forms the tangent vector: Torsion scalarof the manifold.RInstead of the Ricci scalar for the Lagrangian densityin general relativity, the teleparallel Lagrangian densityis described by the torsion scalar .< Modified teleparallel action for f(T) theory >TF(T) ñ T + f(T)

No. 24*・: Gravitational field equation [Bengochea and Ferraro, Phys.A prime denotes a derivative with respect to .We assume the flat FLRW space-timewith the metric.Rev. D 79, 124019 (2009)]Modified Friedmann equations in the flat FLRW background:T,*A prime denotes a derivative with respect to .・ We consider only non-relativistic matter (cold dark matter and baryon).

Combined f(T) theory >No. 25u(> 0):Positiveconstantu =0.5(dash-dottedline)u =0.8(dashed line)LogarithmictermExponentialtermu =1(solid line)w DE = à 1Crossing ofthe phantomdivide・The modelcontains onlyone parameteru if one hasthe valueof .Ω (0)m

IV. Effective equation of state for the universe and thefinite-time future singularities in non-local gravityNo. 26Non-local gravityproduced by quantum effects[Deser and Woodard, Phys. Rev. Lett. 99, 111301 (2007)]・ It is known that so-called matter instability occurs in F(R) gravity.[Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)]This implies that the curvature inside matter sphere becomesvery large and hence the curvature singularity could appear.[Arbuzova and Dolgov, Phys. Lett. B 700, 289 (2011)]It is important to examine whether there exists the curvaturesingularity, i.e., “the finite-time future singularities”in non-local gravity.

A. Non-local gravity< Action >g =det(g ö÷ )f : Some function: Metric tensorNo. 27Λ : Cosmological constantNon-local gravityBy introducing two scalar fields and , we find・By the variation of the action in thefirst expression over ø , we obtain(or )Substituting this equation into the action in thefirst expression, one re-obtains the starting action.: Covariant derivative operator: Covariant d'Alembertian: Matter LagrangianQ: Matter fields

Gravitational field equation >No. 28・ ñ・The variation of the action with respect to gives*0: Energy-momentum tensor of matter(prime) : Derivative with respect toWe assume the flat FLRW space-time with the metric and considerthe case in which the scalar fields ñ and ø only depend on time.Gravitational field equations in the FLRW background:ñ< Equations of motion for ñ and ø >,and :Energy density andpressure of matter.

・In the flat FLRW space-time, we analyze an asymptoticsolution of the gravitational field equations in the limit ofthe time when the finite-time future singularities appear.We consider the case in which the Hubble parameter is expressed ash s : Positive constant,q : Non-zero constant larger than -1When , →∞.t s・No. 29Scale factor :・ We have .ñ c ,a s : Constantø c : Integration constants・We take a form of f(ñ) as .,: Non-zero constants

・It is known that the finite-time future singularities can be classifiedin the following manner:Type I (“Big Rip”):Type II (“sudden”):Type III:Type IV:***In the limit ,[Nojiri, Odintsov and Tsujikawa,Phys. Rev. D 71, 063004 (2005)], ,The case in which ú eff and P eff becomesfinite values at is also included., ,, ,, ,HHigher derivatives of diverge.ú eff |Peff |The case in which and/orasymptotically approach finite values isalso included.No. 30

・The finite-time future singularitiesdescribed by the expression of H in nonlocalgravity have the following properties:No. 31For ,For ,For ,Type I (“Big Rip”)Type II (“sudden”)Type III*f(ñ) HRange and conditions for the value of parameters of , , andand ø c in order that the finite-time future singularities can exist.ñ c

We examine the asymptotic behavior of w eff in the limitby taking the leading term in terms of .No. 32q > 1・ For [Type I (“Big Rip”) singularity],we ff evolves from the non-phantom phase or the phantom oneand asymptotically approaches .w eff = à 1・0 < q < 1For [Type III singularity],evolves from the non-phantom phase to the phantom onewith realizing a crossing of the phantom divide orevolves in the phantom phase.w effà 1 < q < 0The final stage is the eternal phantom phase.・For [Type II (“sudden”) singularity],at the final stage.w eff > 0

・We estimate the present valueof w eff .For case ,*No. 33We regard at the presenttime because the energy density of darkenergy is dominant over that of nonrelativisticmatter at the present time.: The present time has the dimension of: Current value of H, [Freedman et al. [HST Collaboration],Astrophys. J. 553, 47 (2001)]・0 < q < 1For ,.・For à 1 < q < 0 , w eff > 0 .w effw DEIn our models, canhave the presentobserved value of .

V. Summary・ We have discussed modified gravitational theories inorder to explain the current accelerated expansion ofthe universe, so-called dark energy problem.・ We have investigated the equation of state for dark energyin f(R) gravity as well as f(T) theory.w DEw DE = à 1The future crossings of the phantom divide lineare the generic feature in the existingviable f(R) gravity models.No. 34・The crossing of the phantom divide line can berealized in the combined f(T) theory.We have studied the effective equation of state for theuniverse when the finite-time future singularities occurin non-local gravity.

Future evolutions of H as functions of z >No. 23Exponential gravity modelOscillatory behavior:ñH(z=à1)H0‘f’ denotes the valueat the final stagez = à 1.:Present value of theHubble parameter

Hu-Sawicki modelStarobinsky’s modelNo. 24ñH(z=à1)H0Tsujikawa’s modelExponential gravity modelOscillatory behavior:Presentvalue of theHubbleparameter

・・In the future ( ), the crossings ofthe phantom divide are the generic feature for all theexisting viable f(R) models.As z decreases ( ), dark energybecomes much more dominant over non-relativistic matter( ).No. 25< Effective equation of state for the universe >: Total energy density of the universe: Total pressure of the universeP DEP mP r: Pressure of dark energy: Pressure of non-relativistic matter(cold dark matter and baryon): Pressure of radiation

・・Hw eff = à 1 à 32H ç2(a)(b)H ç < 0 w eff > à 1H ç =0w eff = à 1w eff < à 1Non-phantom phaseCrossing of thephantom divide(c) H ç > 0Phantom phaseThe physical reason why the crossing of the phantomdivide appears in the farther future ( )is that the sign of H ç changes from negative topositive due to the dominance of dark energy overnon-relativistic matter.As in the farther future, w DE oscillatesaround the phantom divide line w DE = à 1 becausethe sign of H ç changes and consequently multiplecrossings can be realized.No. 26

Conditions for the viability of f(R) gravity >(1) f 0 (R) > 0f 0 (R) ñ df(R)/dR(2)(3)f 00 (R) > 0f 00 (R) ñ d 2 f(R)/dR 2(4) 0 0 GM 2 ù 1/(3f 00 (R)) > 0[Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)][Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)][Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)][Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)](5) Constraints from the violation of the equivalence principleScale-dependence ‘‘Chameleon mechanism’’M = M(R) :No. 12Positivity of the effective gravitational couplingStability condition:: Gravitational constantM :Mass of a new scalar degree of freedom (“scalaron”)in the weak-field regime.Existence of a matterdominatedstagef(R) → R à 2Λ for R ý R 0 .R 0 : Current curvature, Λ : Cosmological constant Stability of the latetimede Sitter pointCf. For general relativity,m =0.Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)](6) Solar-system constraints [Chiba, Phys. Lett. B 575, 1 (2003)][Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]

Flat Friedmann-Lema tre-Robertson-Walker (FLRW)space-time >a(t): Scale factorNo. 20Gravitational field equations in the FLRW background:: Hubble parameterú M and P M : Energy density and pressure of all perfectfluids of matter, respectively.< Analysis method > [Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)](1)・ Ricci scalar:(2)0(prime): Derivative withrespect toR

We solve Equations (1) and (2) by introducing thefollowing variables:ú mú r: Energy density of non-relativistic matter (cold dark matter and baryon): Energy density of radiationNo. 21ú DE・ ‘(0)’denotes the present values.: Energy density of dark energy(3)(4)

Combining Equations (3) and (4), we obtainNo. 22: Equation fory H

Equation of state for (the component corresponding to) darkenergy >No. 23Pw DE ñ DEúDE< Continuity equation for dark energy >

No. 31: Gravitational field equation*A prime denotes a derivativewith respect to . T[Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)]・We assume the flat FLRW space-time with the metric,,Modified Friedmann equations in the flat FLRW background:*A prime denotes a derivativewith respect to .

No. 32We consider only non-relativistic matter (cold dark matter andbaryon) with and .Continuity equation:・ We define a dimensionlessvariable:: Evolution equation of the universe

Gravitational field equation >No. 36・ The variation of the action with respect to ñ gives0: Energy-momentum tensor of matter(prime): Derivative with respect to< Flat Friedmann-Lema tre-Robertson-Walker (FLRW) metric >a(t): Scale factorñ・We consider the case in which the scalar fieldsdepend on time.ñøand only

Gravitational field equations in the FLRW background:No. 37: Hubble parameterand: Energy density and pressure of matter, respectively.For a perfect fluid of matter:< Equations of motion for ñ and ø >

A. Finite-time future singularitiesNo. 38In the flat FLRW space-time, we analyze an asymptotic solutionof the gravitational field equations in the limit of the time t swhen the finite-time future singularities appear.・We consider the case in which the Hubble parameter is expressed as・When ,Scale factorh s : Positive constantq : Non-zero constant larger than -1We only consider the period .→∞a s : Constant

・By using and ,No. 39・We take a form of f(ñ) as .ñ c : Integration constant,: Non-zero constants・By using and ,ø c : Integration constantThere are three cases.,,,

Other model >・No. 18Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)]f AB (R)= R 2+ 1 2b1log [ cosh(b 1 R) à tanh(b 2 ) sinh(b 1 R) ]b 1 (> 0), b 2: Constant parameters

Future crossing of the phantom divide(i) Hu-Sawicki model(ii) Starobinsky’s model[Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)]p =1c 1 =1 c 2 =1 ,[Starobinsky, JETP Lett. 86, 157 (2007)]n =2õ =1.5No. 30(iii) Tsujikawa’s model[Tsujikawa, Phys. Rev. D 77, 023507 (2008)]ö =1(iv) Exponential gravity model[Cognola, Elizalde, Nojiri, Odintsov,Sebastiani and Zerbini, Phys. Rev. D 77,046009 (2008)][Linder, Phys. Rev. D 80, 123528 (2009)]ì =1.8

・・We examine the behavior of each term of the gravitational fieldequations in the limit , in particular that of the leadingterms, and study the condition that an asymptotic solution canbe obtained.the leading termFor case , ø c =1 vanishes in bothgravitational fieldFor case , equations.No. 16Thus, the expression of the Hubble parameter can be aleading-order solution in terms of for thegravitational field equations in the flat FLRW space-time.This implies that there can exist the finite-timefuture singularities in non-local gravity.

B. Relations between the model parameters and the propertyof the finite-time future singularities・f s and û characterize the theory ofnon-local gravity.h s , t q・s and specify the property ofthe finite-time future singularity.・・ñ cø cand determine a leading-order solution in terms offor the gravitational field equations in the flat FLRW space-time.When ,for ,for and ,for ,H ú effú sfor , asymptotically becomes finite and alsoasymptotically approaches a finite constant value .for , →∞,,No. 17

B. Estimation of the current value of the effective equation of No. 22state parameter for non-local gravity [Komatsu et al. [WMAP Collaboration],・・The limit on a constant equation ofstate for dark energy in a flatuniverse has been estimated asFor a time-dependent equation of statefor dark energy, by using a linear form,constraints on andhave been found as,Astrophys. J. Suppl. 192, 18 (2011)]by combining the data of Seven-Year Wilkinson MicrowaveAnisotropy Probe (WMAP)Observations with the latestdistance measurements from thebaryon acoustic oscillations(BAO) in the distribution ofgalaxies and the Hubble constantmeasurement.: Current value of: Derivative ofw DEw DEfrom the combination of theWMAP data with the BAO data,the Hubble constant measurementand the high-redshift SNe Ia data.

IV. Effective equation of state for the universe andphantom-divide crossingA. Cosmological evolution of the effective equation of state forthe universe・The effective equation ofstate for the universe:No. 20H ç < 0H ç =0: The non-phantom (quintessence) phasew eff > à 1w eff = à 1 Phantom crossing,H ç > 0 : The phantom phasew eff < à 1

(1) General relativistic approach・ Cosmological constant・ Scalar field :[Chiba, Sugiyama and Nakamura, Mon. Not. Roy. Astron. Soc. 289, L5 (1997)][Caldwell, Dave and Steinhardt, Phys. Rev. Lett. 80, 1582 (1998)]K-essence[Chiba, Okabe and Yamaguchi, Phys. Rev. D 62, 023511 (2000)][Armendariz-Picon, Mukhanov and Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)]TachyonX matter, QuintessenceCf. Pioneering work: [Fujii, Phys. Rev. D 26, 2580 (1982)]PhantomNon canonical kinetic termString theories[Padmanabhan, Phys. Rev. D 66, 021301 (2002)]・ Chaplygin gasWrong sign kinetic term[Caldwell, Phys. Lett. B 545, 23 (2002)]p = à A/ú[Kamenshchik, Moschella and Pasquier, Phys. Lett. B 511, 265 (2001)]No. 4Canonical fieldA>0 : Constantú : Energy density: Pressure p

(2) Extension of gravitational theory・ f(R) gravityf(R)・ Scalar-tensor theories・ f(G) gravityCf. Application to inflation:[Starobinsky, Phys. Lett. B 91, 99 (1980)]: Arbitrary function of the Ricci scalar[Capozziello, Cardone, Carloni and Troisi, Int. J. Mod. Phys. D 12, 1969 (2003)][Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, 043528 (2004)][Nojiri and Odintsov, Phys. Rev. D 68, 123512 (2003)]f 1 (þ)RRNo. 5f i (þ) : Arbitrary function(i =1, 2) of a scalar field þ[Boisseau, Esposito-Farese, Polarski and Starobinsky, Phys. Rev. Lett. 85, 2236 (2000)][Gannouji, Polarski, Ranquet and Starobinsky, JCAP 0609, 016 (2006)]・ Ghost condensates[Arkani-Hamed, Cheng, Luty and Mukohyama,JHEP 0405, 074 (2004)]・ Higher-order curvature termf 2 (þ)GGauss-Bonnet term with a coupling to a scalar field:G ñ R 2 à: Ricci curvature[Nojiri, Odintsov and Sasaki, Phys. Rev. D 71, 123509 (2005)] tensorR : Riemann+ f(G)2ô 2ô 2 ñ 8ùGtensor[Nojiri and Odintsov, Phys. Lett. B 631, 1 (2005)] G : Gravitational constant

・ DGP braneworld scenario[Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)][Deffayet, Dvali and Gabadadze, Phys. Rev. D 65, 044023 (2002)]・ f(T) gravityNo. 6: Extended teleparallel Lagrangian density described by thetorsion scalar T .[Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)][Linder, Phys. Rev. D 81, 127301 (2010) [Erratum-ibid. D 82, 109902 (2010)]]・ “Teleparallelism” : One could use the Weitzenböck connection, which has nocurvature but torsion, rather than the curvature defined bythe Levi-Civita connection.[Hayashi and Shirafuji, Phys. Rev. D 19, 3524 (1979) [Addendum-ibid. D 24, 3312 (1982)]]・ Galileon gravity [Nicolis, Rattazzi and Trincherini, Phys. Rev. D 79・, 064036(2009)] Review: [Tsujikawa, Lect. Notes Phys. 800, 99 (2010)]Longitudinal graviton (i.e. a branebending mode þ )The equations of motion are invariant under the Galilean shift:One can keep the equations of motion up to the second-order.This property is welcome to avoid the appearance of an extra degree offreedom associated with ghosts.: Covariant d'Alembertian・ Non-local gravity Quantum effects [Deser and Woodard, Phys.Rev. Lett. 99, 111301 (2007)]

Conditions for the viability of f(R) gravity >(1) f 0 (R) > 0f 00 (R) > 0(2) [Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)](3)・Positivity of the effective gravitational couplingG eff = G 0 /f 0 (R) > 0 G 0 : Gravitational constant(The graviton is not a ghost.)・ Stability condition: M 2 ù 1/(3f 00 (R)) > 0M : Mass of a new scalar degree of freedom (called the“scalaron”) in the weak-field regime.(The scalaron is not a tachyon.)f(R) → R à 2Λ for R ý R 0R 0Λ: Current curvatureΛ : Cosmological constant・ Realization of the CDM-like behavior in the largecurvature regimeNo. 14Standard cosmology [ Λ + Cold dark matter (CDM)]

(4) Solar system constraintsf(R) gravity[Bertotti, Iess and Tortora,Nature 425, 374 (2003).]・Equivalent[Chiba, Phys. Lett. B 575, 1 (2003)]Brans-Dicke theorywith ω BD =0[Erickcek, Smith and Kamionkowski, Phys. Rev. D 74, 121501 (2006)][Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]However, if the mass of the scalar degree of freedom Mis large, namely, the scalar becomes short-ranged, it hasno effect at Solar System scales.M = M(R)‘‘Chameleon mechanism’’・ Scale-dependence:ω BD : Brans-Dicke parameterObservational constraint: |ω BD | > 40000Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)]The scalar degree of freedom may acquire a large effective massat terrestrial and Solar System scales, shielding it fromexperiments performed there.No. 15

(5) Existence of a matter-dominated stage and thatof a late-time cosmic acceleration・ Combing local gravity constraints, it is shown thatm ñ Rf 00 (R)/f 0 (R)has to be several orders ofmagnitude smaller than unity.mquantifies the deviation from the Λ CDM model.[Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)][Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)](6) Stability of the de Sitter spacef d = f(R d )(fd 0)2 à 2f d fd00R dfd 0f > 0d00:・Constant curvaturein the de Sitter spaceLinear stability of the inhomogeneous perturbations inthe de Sitter spaceCf.R d =2f d /f 0 d m

Ω DE Ω m Ω rNo. 19< Cosmological evolutions of , and inthe exponential gravity model >From [KB, Geng and Lee,JCAP 1008, 021 (2010)].f E (R)=ì =1.8

Conclusions of Sec. II >・We have explicitly shown that the future crossingsof the phantom divide are the generic feature in theexisting viable f(R) gravity models.・ We have also illustrated that the cosmological horizonentropy oscillates with time due to the oscillatorybehavior of the Hubble parameter.・ The new cosmological ingredient obtained in thisstudy is that in the future the sign of H ç changesfrom negative to positive due to the dominance ofdark energy over non-relativistic matter.This is a common physical phenomena to the existingviable f(R) models and thus it is one of the peculiarproperties of f(R) gravity models characterizing thedeviation from the Λ CDM model.No. 41

Conclusions of Sec. III >・・・We have investigated the cosmological evolution in the exponentialf(T) theory.The phase of the universe depends on the sign of theparameter p , i.e., for p 0) the universe is always inthe non-phantom (phantom) phase without the crossing ofthe phantom divide.We have presented the logarithmic type f(T) model.It does not allow the crossing of the phantom divide.To realize the crossing of the phantom divide, we haveconstructed an f(T) theory by combining the logarithmic andexponential terms.The crossing in the combined f(T) theory is fromto w DE < à 1 , which is opposite to the typical manner inf(R) gravity models.This combined theory is consistent with the recentobservational data of SNe Ia, BAO and CMB.No. 51w DE > à 1

Conclusions of Sec. IV >No. 64・・We have explicitly shown that three types of the finitetimefuture singularities (Type I, II and III) can occurin non-local gravity and examined their properties.We have investigated the behavior of the effectiveequation of state for the universe when the finitetimefuture singularities occur.

Continuity equation:No. 44・ We define a dimensionlessvariable:: Evolution equation of the universe< (a). Exponential f(T) theory >p : Constant・・p =0The case in which corresponds to the CDM model.This theory contains only one parameter p if the value of Ω (0)mis given.Λ

・We assume the flat FLRW space-time with the metric,A prime denotes a derivativewith respect to .T: Gravitational field equation [Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)]No. 38,Modified Friedmann equations in the flat FLRW background:A prime denotes a derivativewith respect to .We consider only non-relativistic matter (cold dark matter and baryon) withand .

Continuity equation:No. 39・ We define a dimensionlessvariable:: Evolution equation of the universe

No. 45p =0.001p =0.01p = à 0.1p =0.1p = à 0.01・・p>0 p

does not cross the phantom divide linein the exponential f(T) theory.No. 46w DE w DE = à 1・・・For p > 0 , the universe always stays in the non-phantom(quintessence) phase ( w DE > à 1 ), whereas for p < 0it in the phantom phase ( w DE < à 1 ).The larger | p | is, the larger the deviation of theexponential f(T) theory from the Λ CDM model is.We have taken the initial conditions at z =0as.

(b). Logarithmic f(T) theory >No. 47q(> 0): Positive constant・This theory contains onlyone parameter q if the valueof Ω (0) is obtained.mw DEw DE = à 1does not crossthe phantom divideline .

Cosmological evolutions of Ω DE , Ω m and Ω r >No. 41Ω DEΩ rRadiation-dominatedstage [ ]Matter-dominated stageu =1Ω mDark energy becomesdominant over matter( z < 0.36 ).

The best-fit values >No. 42The minimum ( )of the combined f(T) theoryis slightly smaller than thatof the CDM model.Contours of , andconfidence levels in the plane from SNe Ia, BAO andCMB data.The plus sign depicts the best-fit point.Λÿ 2ÿ 2 minThe combined f(T)theory can fit theobservational data well.

Backup Slides A

(4) Solar system constraintsf(R) gravity[Bertotti, Iess and Tortora,Nature 425, 374 (2003).]・Equivalent[Chiba, Phys. Lett. B 575, 1 (2003)]Brans-Dicke theorywith ω BD =0[Erickcek, Smith and Kamionkowski, Phys. Rev. D 74, 121501 (2006)][Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]However, if the mass of the scalar degree of freedom Mis large, namely, the scalar becomes short-ranged, it hasno effect at Solar System scales.M = M(R)‘‘Chameleon mechanism’’・ Scale-dependence:ω BD : Brans-Dicke parameterObservational constraint: |ω BD | > 40000Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)]The scalar degree of freedom may acquire a large effective massat terrestrial and Solar System scales, shielding it fromexperiments performed there.No. 15

(5) Existence of a matter-dominated stage and thatof a late-time cosmic acceleration・ Combing local gravity constraints, it is shown thatm ñ Rf 00 (R)/f 0 (R)has to be several orders ofmagnitude smaller than unity.mquantifies the deviation from the Λ CDM model.[Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)][Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)](6) Stability of the de Sitter spacef d = f(R d )(fd 0)2 à 2f d fd00R dfd 0f > 0d00:・Constant curvaturein the de Sitter spaceLinear stability of the inhomogeneous perturbations inthe de Sitter spaceCf.R d =2f d /f 0 d m

(4) Stability of the late-time de Sitter point0

Data fitting of w(z)No. 22(2) >From [Alam, Sahni and Starobinsky,JCAP 0702, 011 (2007)].SN gold data set+CMB+BAO SNLS data set+CMB+BAO・ ・2û confidence level.

Data fitting of w(z) (3) >No. B-7From [Zhao, Xia, Feng and Zhang,Int. J. Mod. Phys. D 16, 1229 (2007)[arXiv:astro-ph/0603621]]68%Best-fitconfidence level95% confidence level157 “gold” SN Ia data set+WMAP 3-year data+SDSSwith/without dark energy perturbations.

・For most observational probes (except the SNLSdata), a low Ω 0m prior (0.2 < Ω 0m < 0.25) leads toan increased probability (mild trend) for the crossingof the phantom divide.Ω 0m : Current density parameter of matterNo. B-6[Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)]

Bekenstein-Hawking entropy on the apparent horizon inthe flat FLRW background >S = 4GA: Bekenstein-Hawking entropyNo. 30・rà=1/H:: Area of the apparent horizonRadius of the apparent horizon inthe flat FLRW space-timeIt has been shown that it is possible to obtain a picture ofequilibrium thermodynamics on the apparent horizon inthe FLRW background for f(R) gravity due to a suitableredefinition of an energy momentum tensor of the “dark”component that respects a local energy conservation.[KB, Geng and Tsujikawa, Phys. Lett. B 688, 101 (2010)]In this picture, the horizonà áentropy is simplyexpressed as S = ù/ GH 2 .

Future evolutions of S as functions of z >No. 36Exponential gravity modelñS(z=à1)S0Oscillating behavior:Present value of thehorizon entropy

Hu-Sawicki modelStarobinsky’s modelNo. 37ñS(z=à1)S0Tsujikawa’s modelExponential gravity modelOscillating behavior: Presentvalue of thehorizonentropy

S ∝ H à2H・ Since , the oscillating behavior of comesfrom that of .Cf.However, it should be emphasized that althoughS decreases in some regions, the second law ofthermodynamics in f(R) gravity can be alwayssatisfied.SThis is because is the cosmological horizonentropy and it is not the total entropy of theuniverse including the entropy of generic matter.It has been shown that the second law ofthermodynamics can be verified in both phantom andnon-phantom phases for the same temperature of theuniverse outside and inside the apparent horizon.S[KB and Geng, JCAP 1006, 014 (2010)]No. 40

(a). Exponential f(T) theory >No. 16

No. 16< (b). Logarithmic f(T) theory >

(c). Combined f(T) theory >No. 16

(c). Combined f(T) theory >No. 16

p : Constantú Mand P M : Energy density and pressureof all perfect fluids ofgeneric matter, respectively.・Initial conditions:

Combined f(T) theory >No. 48u : Constant( )u =0.5(dash-dottedline)u =0.8(dashed line)LogarithmictermExponentialtermu =1(solid line)w DE = à 1Crossing ofthe phantomdivide・The modelcontains onlyone parameteru if one hasthe valueof .Ω (0)m

IV. Effective equation of state for the universe and thefinite-time future singularities in non-local gravity・Non-local gravityproduced by quantum effects[Deser and Woodard, Phys. Rev. Lett. 99, 111301 (2007)]There was a proposal on the solution of the cosmologicalconstant problem by non-local modification of gravity.[Arkani-Hamed, Dimopoulos, Dvali and Gabadadze, arXiv:hep-th/0209227]Recently, an explicit mechanism to screen a cosmologicalconstant in non-local gravity has been discussed.[Nojiri, Odintsov, Sasaki and Zhang, Phys. Lett. B 696, 278 (2011)]Recent related reference: [Zhang and Sasaki, arXiv:1108.2112 [gr-qc]]・ It is known that so-called matter instability occurs in F(R) gravity.[Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)]This implies that the curvature inside matter sphere becomesvery large and hence the curvature singularity could appear.It is important to examine whether there exists the curvaturesingularity, i.e., “the finite-time future singularities”in non-local gravity.No. 51[Arbuzova and Dolgov, Phys. Lett. B 700, 289 (2011)]

C. Relations between the model parameters and the propertyof the finite-time future singularities・・・・ñ cø cf s and û characterize the theory ofnon-local gravity.h sq, t s and specify the property ofthe finite-time future singularity.and determine a leading-order solution in terms offor the gravitational field equations in the flat FLRW space-time.When ,for ,for , ,for and ,No. 59,H ú effú sfor , asymptotically becomes finite and alsoasymptotically approaches a finite constant value .for , →∞,

・It is known that the finite-time future singularities can be classifiedin the following manner:In the limit ,Type I (“Big Rip”): , ,* The case in which ú eff and P eff becomesfinite values at is also included.Type II (“sudden”):Type III:Type IV:[Nojiri, Odintsov andTsujikawa, Phys. Rev.D 71, 063004 (2005)]**, ,, ,, ,HHigher derivatives of diverge.ú eff |Peff |The case in which and/orasymptotically approach finite values isalso included.No. 60

Appendix A

Other models >No. A-10・Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)]f AB (R)= R 2+ 1 2b1log [ cosh(b 1 R) à tanh(b 2 ) sinh(b 1 R) ]b 1 (> 0), b 2: Constant parametersCf. Power-law modelf(R)=R à öR vö(> 0): Constant parameter[Amendola, Gannouji, Polarski and Tsujikawa,Phys. Rev. D 75, 083504 (2007)][Li and Barrow, Phys. Rev. D 75, 084010 (2007)]0

S = 4GA:Bekenstein-Hawking horizon entropyin the Einstein gravityNo. A-11・・:Wald entropy in modified gravitytheories: Area of the apparent horizonWald introduced a horizon entropy associated with aNoether charge in the context of modified gravity theories.This is equivalent to S ê = A/(4G eff ).S êThe Wald entropy is a local quantity defined in terms ofquantities on the bifurcate Killing horizon.More specifically, it depends on the variation of theLagrangian density of gravitational theories withrespect to the Riemann tensor.G eff = G/FS ê: Effective gravitational coupling

w effNo. A-13< Cosmological evolution of in the exponentialgravity model >From [KB, Geng and Lee,JCAP 1008, 021 (2010)].f E (R)=

Remarks >(a) The qualitative results do not strongly depend on thevalues of the parameters in each model.(b) The evolutions of the Wald entropy are similar toin the models of (i)‐(iv).S = 4GA:Bekenstein-Hawking horizon entropyin the Einstein gravityS êNo. A-14Cf. [KB, Geng and Tsujikawa, Phys. Lett. B 688, 101 (2010)]S[KB, Geng and Lee, JCAP 1008, 021 (2010)]S ê =F(R)A4G:Wald entropy in modified gravitytheories including f(R) gravity(c)The numerical results in the Appleby-Battye modelare similar to those in the Starobinsky model of (ii).

Cosmological evolutions of , and in theexponential gravity model >S ö S êö H ö No. A-15From [KB, Geng and Lee,JCAP 1008, 021 (2010)].f E (R)=

Numerical results >Models of (i), (ii), (iii) and (iv)・ w DE (z =0)=・・・=(-0.76, -0.82), (-0.83, -0.98), (-0.79, -0.80) and (-0.74, -0.80)z crossz p::Value of z at the first future crossing of thephantom dividezValue of at the first sign change of fromnegative to positive=, ,,andH çNo. A-16

Initial conditions >Models of (i), (ii), (iii) and (iv)・We have taken the initial conditions at z = z 0 , so thatwith, to ensurethat they can be all close enough to the Λ CDM modelwith .In order to save the calculation time, the different valuesof z 0 mainly reflect the forms of the models, i.e., thepower-law types of (i) and (ii) and the exponential onesof (iii) and (iv).・ At z = z 0 , w DE = à 1 .No. A-19

No. A-20・Models of (i), (ii), (iii) and (iv)= , ,and・[E. Komatsu et al. [WMAP Collaboration], Astrophys. J.Suppl. 192, 18 (2011) [arXiv:1001.4538 [astro-ph.CO]]]In the high z regime ( ), , inwhich f(R) gravity has to be very close to the ΛCDMmodel.・・

No. A-21・・ By examining the cosmological evolutions of andw DE as functions of the redshift z for the models, wehave found that is close to its initial valueof .y H・This is because in the higher z regime, the universe is inthe phantom phase ( w DE < à 1) and therefore, ú DE andy H increase (since ), whereas in the lower zregime, the universe is in the non-phantom (quintessence)phase ( w DE > à 1 ) and hence they decrease.Consequently, the above two effects cancel out.

・Our results are not sensitive to the initial values ofand .No. A-22z 0・The initial condition ofisdue to that the f(R) gravity models at z = z 0 should bevery close to the ΛCDM model, in which.

No. A-23

Second law of thermodynamics in equilibriumdescription >We consider the same temperature of the universeoutside and inside the apparent horizon.< Gibbs equation >No. A-17[KB and Geng, JCAP 1006, 014 (2010)]< Second law of thermodynamics >< Condition >As long as R>0 , the second law of thermodynamicscan be met in both non-phantom ( H ç < 0 , w eff > à 1 )and phantom ( H ç > 0, w eff < à 1 ) phases.Cf. [Gong, Wang and Wang, JCAP 0701, 024 (2007)][Jamil, Saridakis and Setare, Phys. Rev. D 81, 023007 (2010)]

Second law of thermodynamics >[KB and Geng, JCAP1006, 014 (2010)]No. A-18We assume the same temperature between the outsideand inside of the apparent horizon.< Gibbs equation > : Entropy of total energy inside the horizon< Second law of thermodynamics >< Condition >[Wu, Wang, Yang and Zhang, Class.Quant. Grav. 25, 235018 (2008)]F > 0 because G eff = G/F > 0 .The second law of thermodynamics in f(R) gravitycan be satisfied in phantom ( , )as well as non-phantom ( H ç H ç > 0 w eff < à 1< 0 , w eff > à 1 ) phases.: Effective equation of state (EoS)

Backup Slides B

Residuals for the best fit to a flat Λ cosmology >Δ(m à M)No. BS-B1Flat ΛcosmologyzPure mattercosmologyFrom [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]

Baryon acoustic oscillation (BAO) >No. BS-B2Pure CDM model(No peak)Ω m h 2 =0.12, 0.13, 0.14, 0.105(From top to bottom)Ω b h 2 =0.024From [Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)]

5-year WMAP data on >・ For the flat universe, constant :à 0.14 < 1+w

・No. BS-B4In the flat FLRW background, gravitational field equations read,ú eff , p eff: Effective energydensity andpressure from theterm f(R) à R・ Example : f(R) ∝ R n (n6=1)à2n 2 +3nà1a ∝ t q , q = nà2w eff = à 6n 2 à9n+3q > 16n 2 à7nà1[Capozziello, Carloni andTroisi, Recent Res. Dev.Astron. Astrophys. 1, 625(2003)]If , accelerated expansion can be realized.(For n =3/2 or n = à 1 , q =2 and w eff = à 2/3 .)

Appendix B

(7) Free of curvature singularities・ Existence of relativistic stars[Frolov, Phys. Rev. Lett. 101, 061103 (2008)]No. B-2[Kobayashi and Maeda, Phys. Rev. D 78, 064019 (2008)][Kobayashi and Maeda, Phys. Rev. D 79, 024009 (2009)]< Model with satisfying the condition (7) >F MJWQ (R)ë>0, R ã > 0 : Constants[Miranda, Joras, Waga and Quartin, Phys. Rev. Lett. 102, 221101 (2009)]Cf. [de la Cruz-Dombriz, Dobado and Maroto, Phys. Rev. Lett. 103, 179001 (2009)]

Recent work >No. B-3[Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, 123516 (2009)]Cf. [Nozari and Azizi, Phys. Lett. B 680, 205 (2009)]・It has been shown that in the Hu-Sawicki model, the transitionfrom the phantom phase to the non-phamtom one can also occur.From [Martinelli,Melchiorri andAmendola, Phys. Rev.D 79, 123516 (2009)]n =1F 0 (R 0 )=à 0.01F 0 (R 0 )=à 0.03F 0 (R 0 )=à 0.1Ω m =0.24

We reconstruct an explicit model of F(R) gravitywith realizing the crossing of the phantom divide.No. B-4< Preceding work >From [Amendola and Tsujikawa,Phys. Lett. B 660, 125 (2008)]F(R)=(R 1/c à Λ) c・ Example:c, Λ : Constantsc =1.8Phantom phaseNon-phantom phase

(5) Existence of a matter-dominated stage and thatof a late-time cosmic accelerationAnalysis of m(r) curve on the (r, m) planem ñ RF 00 (R)/F 0 (R), r ñàRF 0 (R)/F(R)・ Presence of a matter-dominated stageandatm(r) ù +0dm > à 1dr・ Presence of a late-time acceleration(i)(ii)・√3m(r) =à r à 1, à10, p: ConstantsNo. B-5[Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)][Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)]